1
• Position, Velocity, Acceleration
• Significant Figures, Measurements, Errors
• Vectors, Relative Motion
• Studying Tips
• Equilibrium and Non-equilibrium Problems
• Circular Motion, Centripetal Force
1
PHY131 1st half Exam
Jam
The Particle Model
Often motion of the object as a whole is not influenced by
details of the object’s size and shape
We only need to keep track of a single point on the object
So we can treat the object as if all its mass were
concentrated into a single point
A mass at a single point in space is called a particle
Particles have no size, no shape and no top, bottom, front
or back
Below us a motion diagram of a car stopping, using the
particle model
2
• Below is a motion diagram, made at 1 frame per minute, of a student walking to school.
A motion diagram is one way to represent the student’s motion.
Another way is to make a graph of x versus t for the student:
Position-versus-Time Graphs
3
Two balls are launched along a pair of tracks with equal
velocities, as shown. Both balls reach the end of the
track. Predict: Which ball will reach the end of the track
first?
• A
• B
• C: They will reach the end of the track at the same
time
Demo: Two balls were launched along a pair of tracks
with equal velocities. Both balls reached the end of the
track. Observe: Which ball reached the end of the track
first?
• A
• B
• C: They reached the end of the track at the same
time
Explanation: Why does ball B reach the end of the track first? A. Ball B is always traveling faster than ball A, so it reaches the end
of the track first.
B. Balls A and B start and end with the same speed. But while ball B is on the lower part, it is going faster than ball A because gravity has sped it up. Its average speed is greater, so it gets there first.
C. Ball B travels a shorter distance than ball A.
D. Ball B travels a longer distance, but is pulled faster by an extra force we cannot know about.
E. The observation is flawed – ball B should not reach the end first.
2
Acceleration
Sometimes an object’s velocity
is constant as it moves
More often, an object’s velocity
changes as it moves
Acceleration describes a change
in velocity
Consider an object whose velocity changes from v1 to v2
during the time interval Δt
The quantity Δv = v2 – v1 is the change in velocity
The rate of change of velocity is called the average
acceleration:
7
Acceleration (a.k.a. “instantaneous
acceleration”)
dt
vd
t
va
t
0lim
0v
1v
v
Units of
are m/s.
v
a Units of
are m/s2. a
8
A particle has velocity as it accelerates from 1 to 2.
What is its velocity vector as it moves away from
point 2 on its way to point 3?
Unit Conversions
It is important to be able to
convert back and forth between
SI units and other units
One effective method is to write
the conversion factor as a ratio
equal to one
Because multiplying by 1 does not change a value, these
ratios are easily used for unit conversions
For example, to convert the length 2.00 feet to meters, use
the ratio:
So that:
10
Suggested Problem Solving Strategy
• MODEL
• VISUALIZE
• SOLVE
• ASSESS
Think about and simplify the situation,
guess at what the right answer might be.
Draw a diagram. It doesn’t have to be
artistic: stick figures and blobs are okay!
Set up the equations, solve for what
you want to find. (This takes time..)
Check your units, significant figures, do a
“sanity check”: does my answer make sense?
This is just a suggested strategy. Whatever method works for you is fine, as long as you don’t make a mistake, and you show how you got to the correct answer, it’s 100%! 11
Error Analysis
Almost every time you make a measurement, the
result will not be an exact number, but it will be a
range of possible values.
The range of values associated with a measurement
is described by the uncertainty, or error.
Exactly 3 apples (no error)
1600 ± 100 apples:
1600 is the value
100 is the error
12
3
A histogram of many, many
measurements of the same
thing:
N
Errors
Errors eliminate the need to report measurements with
vague terms like “approximately” or “≈”.
Errors give a quantitative way of stating your confidence
level in your measurement.
Saying the answer is 10 ± 2 means you are 68% confident
that the actual number is between 8 and 12.
It also implies that and 14 (the 2-σ range).
±1 σ
±2 σ 13
Suppose you make N measurements of a quantity x, and
you expect these measurements to be normally distributed
Each measurement, or trial, you label with a number i,
where i = 1, 2, 3, etc
You do not know what the true mean of the distribution
is, and you cannot know this
However, you can estimate the mean by adding up all the
individual measurements and dividing by N:
Estimating the Mean from a Sample
xest 1
Nxi
i1
N
14
Suppose you make N measurements of a quantity x, and
you expect these measurements to be normally distributed
It is impossible to know the true standard deviation of the
distribution
The best estimate of the standard deviation is:
Estimating the Standard Deviation from a
Sample
est 1
N 1(xi xest )
2
i1
N
The quantity N – 1 is called the number of degrees of
freedom
In this case, it is the number of measurements minus one
because you used one number from a previous calculation
(mean) in order to find the standard deviation. 15
Reading Error (Digital)
For a measurement with an instrument with a
digital readout, the reading error is usually “±
one-half of the last digit.”
This means one-half of the power of ten
represented in the last digit.
With the digital thermometer shown, the last
digit represents values of a tenth of a degree,
so the reading error is ½ × 0.1 = 0.05°C
You should write the temperature as 12.80 ± 0.05 °C.
16
Propagation of Errors
• Rule #1 (sum or difference rule):
• If z = x + y
• or z = x – y
• then
• Rule #2 (product or division rule):
• If z = xy
• or z = x/y
• then
z x2 y2
z
z
x
x
2
y
y
2
17
Propagation of Errors
• Rule #2.1 (multiply by exact constant rule):
• If z = xy or z = x/y
• and x is an exact number, so that Δx=0
• then
• Rule #3 (exponent rule):
• If z = xn
• then
z x y
z
z n
x
x 18
4
The Error in the Mean
Many individual, independent measurements
are repeated N times
Each individual measurement has the same
error Δx
Using error propagation you can show that the
error in the estimated mean is:
x est x
N19
Free Fall
The motion of an object moving
under the influence of gravity only,
and no other forces, is called free
fall
Two objects dropped from the
same height will, if air resistance
can be neglected, hit the ground at
the same time and with the same
speed
Consequently, any two objects in
free fall, regardless of their mass,
have the same acceleration:
20 The apple and feather seen
here are falling in a vacuum.
Free Fall
Figure (a) shows the motion
diagram of an object that was
released from rest and falls freely
Figure (b) shows the object’s
velocity graph
The velocity graph is a straight
line with a slope:
where g is a positive number which
is equal to 9.80 m/s2 on the surface
of the earth
Other planets have different values
of g 21
Two-Dimensional Kinematics
If the velocity vector’s
angle θ is measured from
the positive x-direction, the
velocity components are
where the particle’s speed
is
Conversely, if we know the velocity components, we can
determine the direction of motion:
22
Projectile Motion
The start of a
projectile’s motion is
called the launch
The angle θ of the
initial velocity v0 above
the x-axis is called the
launch angle
where v0 is the initial speed
The initial velocity vector can be broken into
components
23
Projectile Motion
Gravity acts downward
Therefore, a projectile
has no horizontal
acceleration
Thus
The vertical component of acceleration ay is −g of free fall
The horizontal component of ax is zero
Projectiles are in free fall
24
5
Reasoning About Projectile Motion
If air resistance is neglected,
the balls hit the ground
simultaneously
The initial horizontal velocity of
the first ball has no influence
over its vertical motion
Neither ball has any initial
vertical motion, so both fall
distance h in the same amount
of time
A heavy ball is launched exactly horizontally at height h above
a horizontal field. At the exact instant that the ball is launched, a
second ball is simply dropped from height h. Which ball hits the
ground first?
25
Range of a Projectile
This distance is
sometimes called the
range of a projectile
Example 4.5 from your
textbook shows:
A projectile with initial speed v0 has a launch angle of θ above
the horizontal. How far does it travel over level ground before
it returns to the same elevation from which it was launched?
The maximum distance
occurs for θ = 45°
Trajectories of a projectile launched at
different angles with a speed of 99 m/s.
26
• Relative velocities are found as the time derivative of the relative positions.
• CA is the velocity of C relative to A.
• CB is the velocity of C relative to B.
• AB is the velocity of reference frame A relative to reference frame B.
• This is known as the Galilean transformation of velocity.
Reference Frames
Slide 4-69
Relative Velocity
27
Relative Motion
• Note the “cancellation”
• vTG = velocity of the
Train relative to the
Ground
• vPT = velocity of the
Passenger relative to the
Train
• vPG = velocity of the
Passenger relative to the
Ground
vPG = vPT + vTG
Inner subscripts
disappear 28
What is a force?
A force is a push or a pull
A force acts on an object
Pushes and pulls are applied to something
From the object’s perspective, it has a force exerted on it
• The S.I. unit of force is
the Newton (N)
• 1 N = 1 kg m s–2
29
Tactics: Drawing force vectors
30
6
31
Equilibrium
An object on which the net force is zero is in equilibrium
If the object is at rest, it is in static equilibrium
If the object is moving along a straight line with a constant
velocity it is in dynamic equilibrium
The requirement for either
type of equilibrium is:
The concept of equilibrium is
essential for the engineering
analysis of stationary objects such
as bridges. 32
A ring, seen from above, is pulled on
by three forces. The ring is not moving.
How big is the force F?
A. 20 N
B. 10cos N
C. 10sin N
D. 20cos N
E. 20sin N
Why are you at University?
A. To get a pretty degree with a red
sticker, which I can frame and hang
on my wall
B. My parents said I “have to”
C. You can’t get a good job without a
university education
D. I’m just here to learn interesting
stuff
E. …other
Time Management
• Having a daily and weekly schedule and sticking to it
will improve your marks.
• Organize your time so that “all-nighters” never
happen!
• Most nights you should get an adequate amount of
sleep. An adequate amount of sleep is such that you
do not feel sleepy during the rest of the day.
Food and Exercise
• There are four food groups:
– Vegetables and Fruit
– Grain Products
– Dairy
– Meat and Alternatives (like nuts, tofu, eggs)
• Physical activity not only improves
health but it improves circulation of
blood to the brain.
• Try to get 35-40 minutes of brisk
physical activity, 5 or 6 times per
week. (I ♥ DDR!)
7
Study Groups – working
with Peers
• Find student (students) in class
that you work well with on
MasteringPhysics, end-of-
chapter suggested problems,
and past tests.
• The best way to learn is to teach! If
you can’t explain to someone else
what you have done, you haven’t
really understood it! (This is harder
than you think!)
Past Tests and Exams
• The purpose of obtaining and going through old tests and exams is to get to know “the system”.
• Each course and prof will have a certain pattern and style. Knowing the pattern in advance gives you an edge.
• Don’t count on lightning to strike twice – memorizing old test questions rarely works!
Wed. Dec. 12 evening: Go see a movie!
• The evening before a test is
NOT the best time to study (it
is just the most popular)
• Don’t worry – you have been
studying since the 1st week of
classes!
• On Thursday before 2:00, if
you have time, it can be good
to spend some extra time
reviewing (utilizing short
term memory)
During the Exam
• Skim over the entire exam from front to
back before you begin. Look for problems
that you have confidence to solve first.
• If you start a problem but can’t finish it,
leave it, make a mark on the edge of the
paper beside it, and come back to it after
you have solved all the easy problems.
• When you are in a hurry and your hand is
not steady, you can make little mistakes; if
there is time, do the calculation twice and
obtain agreement.
• Bring a snack or drink.
• Don’t leave a test early!
Suppose the x- and y-components of acceleration are
independent of each other
That is, ax does not depend on y or vy, and ay does not
depend on x or vx
Your problem-solving strategy is to:
1. Draw a free-body diagram
2. Use Newton’s second law in component form:
The force components (including proper signs) are
found from the free-body diagram
Non-Equilibrium
41
Gravity: FG = mg is just a short form!
and
are the same equation, with different notation!
The only difference is that in the second equation
we have assumed that m2 = M (mass of the
earth) and r ≈ R (radius of the earth). 42
8
Weight: A Measurement
The figure shows a man weighing
himself in an accelerating elevator
Looking at the free-body diagram,
the y-component of Newton’s
second law is:
The man’s weight as he
accelerates vertically is:
You weigh more as an elevator
accelerates upward! 43
A. Her weight is the same and her mass is less.
B. Her weight is less and her mass is less.
C. Her weight is less and her mass is the same.
D. Her weight is the same and her mass is the same.
E. Her weight is zero and her mass is the same.
QuickCheck 6.7
An astronaut takes her bathroom scales to the moon,
where g = 1.6 m/s2. On the moon, compared to at
home on earth:
Slide 6-60
Normal Force
When an object sits on a table,
the table surface exerts an
upward contact force on the
object
This pushing force is directed
perpendicular to the surface,
and thus is called the normal
force
A table is made of atoms
joined together by molecular
bonds which can be modeled as
springs
Normal force is a result of
many molecular springs being
compressed ever so slightly 45
Tension Force
When a string or rope or wire
pulls on an object, it exerts a
contact force called the tension
force
The tension force is in the
direction of the string or rope
A rope is made of atoms joined
together by molecular bonds
Molecular bonds can be
modeled as tiny springs holding
the atoms together
Tension is a result of many
molecular springs stretching
ever so slightly
Why does
friction exist?
Because at the
microscopic
level, nothing
is smooth!
A Model of Friction The friction force response to an increasing applied force.
48
9
Rolling Motion
If you slam on the brakes
so hard that the car tires
slide against the road
surface, this is kinetic friction
Under normal driving
conditions, the portion of the
rolling wheel that contacts
the surface is stationary, not
sliding
If your car is accelerating or decelerating or turning,
it is static friction of the road on the wheels that
provides the net force which accelerates the car
49
Drag
The air exerts a drag force on objects as they move
through the air
Faster objects experience a greater drag force than
slower objects
The drag force on a
high-speed
motorcyclist is
significant
The drag force
direction is opposite
the object’s velocity
50
Drag
For normal sized objects on earth traveling at a speed v which
is less than a few hundred meters per second, air resistance can
be modeled as:
A is the cross-section area of the object
ρ is the density of the air, which is about 1.2 kg/m3
C is the drag coefficient, which is a dimensionless number
that depends on the shape of the object
51
Cross Sectional Area depends on size, shape, and
direction of motion.
…Consider the forces on a falling piece of paper,
crumpled and not crumpled. 52
Terminal Speed
The drag force from the air
increases as an object falls and
gains speed
If the object falls far enough, it
will eventually reach a speed at
which D = FG
At this speed, the net force is
zero, so the object falls at a
constant speed, called the
terminal speed vterm
53
Terminal Speed
The figure shows the
velocity-versus-time graph
of a falling object with and
without drag
Without drag, the velocity
graph is a straight line with
ay = −g
When drag is included, the
vertical component of the
velocity asymptotically
approaches −vterm
54
10
Propulsion
If you try to walk across a
frictionless floor, your foot slips
and slides backward
In order to walk, your foot
must stick to the floor as you
straighten your leg, moving
your body forward
The force that prevents
slipping is static friction
The static friction force points
in the forward direction
It is static friction that propels
you forward! What force causes this sprinter
to accelerate? 55
Acceleration Constraints
If two objects A and B move together, their accelerations are
constrained to be equal: aA = aB
This equation is called an acceleration constraint
Consider a car being towed by a truck
In this case, the
acceleration constraint is
aCx = aTx = ax
Because the
accelerations of both
objects are equal, we can
drop the subscripts C and
T and call both of them
ax 56
Acceleration Constraints
Sometimes the acceleration
of A and B may have
different signs
Consider the blocks A and
B in the figure
The string constrains the
two objects to accelerate
together
But, as A moves to the right in the +x direction, B moves
down in the −y direction
In this case, the acceleration constraint is aAx = −aBy
57
The Massless String Approximation
Often in physics problems the mass of the string or rope is
much less than the masses of the objects that it connects.
In such cases, we can adopt the following massless string
approximation:
58
A car is parked on a flat surface.
The car has a mass, m, and a downward force of gravity
on it of magnitude FG = mg.
Why is the normal force equal to mg?
A. Because that is the equation for normal force: n = mg
B. Because acceleration is zero, so the forces must
balance
C. Because of Newton’s 3rd Law: the two forces must
be equal and opposite